Questions on the symmetry of of the alexander polynomial and the rank of the Seifert matrix

Question

For my bachelor thesis I am using the book "Lectures on the topology of 3-manifolds. an introduction to the Casson invariant"(1999) by Nikolai Saveliev.

Regarding the Alexander Polynomial as defined below I have two questions.

1. Why does the equation hold for rank $S$ and not only for $n$ with $n$ the size of the Seifert matrix $S$?

2. Why is the rank of the Seifert matrix $S$ $2g$ with $g$ the genus of the corresponding Seifert surface?

I know Seifert surfaces of genus $g$ are isotopic to a surface in disc-band form with $2g$ bands. Therefore the corresponding Seifert matrix is a $2g\times 2g$ matrix. But this matrix doesn't need to have full rank.

Thanks in advance.

Alexander polynomial definition

Answer 1

The author probably had in mind "dimension" instead of "rank", so the sentence simply means that $S$ is a $2g \times 2g$ matrix since $k$ is a knot (if it was a $n$-component link it would have dimension $2g+n-1$).

In particular, the determinant of a Seifert matrix can very well be zero, eg if $A$ is the Seifert matrix associated to a Seifert surface $F$ for a knot, then the Seifert surface obtained from $F$ by attaching a 1-handle has as associated matrix an enlargement of $A$ with a column/row of zeroes. So the usual "rank" of a matrix doesn't really work here.